逆高斯曲率流展开曲面

平均凸锥在中曲率流下的稳定性

Stability of mean convex cones under mean curvature flow

我们考虑图形解决方案，中曲率流和获得的稳定性结果的homothetically扩大解决方案出来的锥的正中曲率：如果另一个解决方案是最初接近锥在无穷远，然后差异的homothetically扩大解决方案变得很小的大时代。证明涉及建立适当的障碍。

On the spectrum and index of expanding and translating solitons of the mean curvature flow in $\mathbb{R}^3$

In this paper we prove that two-dimensional translating solitons in $\mathbb{R}^3$ with finite $L$-index are homeomorphic to a plane or a cylinder and that a two-dimensional self-expander with finite $L$-index and sub exponential weighted volume growth has finite topology. We also prove that translating solitons and self-expanders have finite topology, provided the bottom of the spectrum of the $L$-stability operator is bounded from below and their weighted volume have subexponential growth.

Classification of expanding and steady Ricci solitons with integral curvature decay

In this paper we prove new classification results for nonnegatively curved gradient expanding and steady Ricci solitons in dimension three and above, under suitable integral assumptions on the scalar curvature of the underlying Riemannian manifold. In particular we show that the only complete expanding solitons with nonnegative sectional curvature and integrable scalar curvature are quotients of the Gaussian soliton, while in the steady case we prove rigidity results under sharp integral scalar curvature decay. As a corollary, we obtain that the only three dimensional steady solitons with less…

Expanding solitons to the Hermitian curvature flow on complex Lie groups

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. Furthermore, we give a structural result concerning expanding semi-algebraic solitons on complex Lie groups. It turns out that the restriction of the soliton metric to the nilradical is also an expanding algebraic soliton and we explain how to construct expanding solitons on comp…

Global existence of Dirac-wave maps with curvature term on expanding spacetimes

We prove the global existence of Dirac-wave maps with curvature term with small initial data on globally hyperbolic manifolds of arbitrary dimension which satisfy a suitable growth condition. In addition, we also prove a global existence result for wave maps under similar assumptions.

A class of anisotropic expanding curvature flows

We consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_k^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_k is the k-th symmetric polynomial of the principle curvature radii of the hypersurface, k is an integer and 1<= k<= n. We prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a soliton which is a solution of a elliptic equation, when the constants alpha, beta bel…

Expanding solitons with non-negative curvature operator coming out of cones

We consider Ricci flow of complete Riemannian manifolds which have bounded non-negative curvature operator, non-zero asymptotic volume ratio and no boundary. We prove scale invariant estimates for these solutions. Using these estimates, we show that there is a limit solution, obtained by scaling down this solution at a fixed point in space. This limit solution is an expanding soliton coming out of the asymptotic cone at infinity.

渐近锥形梯度扩展里奇溶剂的曲率捏紧

Curvature pinching of asymptotically conical gradient expanding Ricci solitons

本文研究了完整非紧致渐近锥形梯度扩展里奇溶剂的曲率捏紧现象，建立了若干Hamilton-Ivey型曲率捏紧估计，包括三维梯度里奇扩展体的正截面曲率结果。

非紧完全Ricci扩展孤子的注记

Remarks on non-compact complete Ricci expanding solitons

本文研究了满足$$ Rc=cg+D^2f, $$的梯度Ricci扩展孤子$(X,g)$，其中$Rc$是Ricci曲率，$c<0$是常数，$D^2f$是$X$上势函数$f$的Hessian.证明了对于具有非负Ricci曲率的梯度扩展孤子$(X,g)$，标量曲率$R$在$X$上至少有一个极大点，它是势函数$f$的唯一极小点.此外，除非$(X,g)$是Ricci平坦的，否则$R>0$在$X$上。我们还证明了$ε$箍缩完全非紧膨胀孤子的标量曲率存在指数衰减。

球面中的曲率流

Curvature flows in the sphere

本文研究球面中严格凸超曲面的收缩与扩张曲率流，利用高斯映射建立两者关系，并证明重缩放后超曲面分别指数收敛到以对应点为中心的单位球面。

Structure at infinity of expanding gradient Ricci soliton

We study the geometry at infinity of expanding gradient Ricci solitons of dimension greater than two with finite asymptotic curvature ratio without curvature sign assumptions. We mainly prove that they have a cone structure at infinity.

高维稳态和扩展溶iton的曲率估计

Curvature estimates for steady and expanding solitons in higher dimensions

本文研究了高维稳态和扩展溶iton的曲率估计，改进了之前的结果并去除了非负里奇曲率假设，同时证明了相关几何结构的存在性和唯一性。

A no expanding breather theorem for noncompact Ricci flows

In this note we show that, under certain curvature positivity conditions (the weak $\operatorname{PIC}-2$ condition or the nonnegative bisectional curvature condition), a complete and noncompact expanding breather of the Ricci flow is also an expanding gradient Ricci soliton. This is the first no expanding breather theorem in the noncompact category.

Strong spherical rigidity of ancient solutions of expansive curvature flows

We consider geometric flows of hypersurfaces expanding by a function of the extrinsic curvature and we show that the homothethic sphere is the unique solution of the flow which converges to a point at the initial time. The result does not require assumptions on the speed other than positivity and monotonicity and it is proved using a reflection argument. Our theorem shows that expanding flows exhibit stronger spherical rigidity, if compared with the classification results of ancient solutions in the contractive case.

具有有限渐近标量曲率比的完备梯度扩张Ricci孤立子

Complete gradient expanding Ricci solitons with finite asymptotic scalar curvature ratio

本文研究维数不小于5、Ricci曲率非负的完备梯度扩张Ricci孤立子，证明若其渐近标量曲率比有限，则黎曼曲率张量至少满足次二次衰减。

四维完备梯度扩张Ricci孤立子的曲率估计

Curvature estimates for four-dimensional complete gradient expanding Ricci solitons

本文研究四维完备非负Ricci曲率梯度扩张Ricci孤立子，得到曲率张量及其协变导数由标量曲率界定的估计式，并应用于证明有限渐近标量曲率比蕴含有限渐近曲率比与$C^{1,α}$渐近锥。

Decay of the Weyl curvature in expanding black hole cosmologies

This paper is motivated by the non-linear stability problem for the expanding region of Kerr de Sitter cosmologies in the context of Einstein's equations with positive cosmological constant. We show that under dynamically realistic assumptions the conformal Weyl curvature of the spacetime decays towards future null infinity. More precisely we establish decay estimates for Weyl fields which are (i) uniform (with respect to a global time function) (ii) optimal (with respect to the rate) and (iii) consistent with a global existence proof (in terms of regularity). The proof relies on a geometric p…

平凡性、旋转对称性与完整扩张梯度 Yamabe 孤子的分类

Triviality, Rotational Symmetry, and Classification of Complete Expanding Gradient Yamabe Solitons

本文研究了完整扩张梯度 Yamabe 孤子的标量曲率，分类了非平凡解，探讨了标量曲率大于或小于孤子常数两种情况下的性质。

R^{n+1} 中平均曲率流的 2-凸平移孤立子与扩张孤立子的凸性

Convexity of 2-convex translating and expanding solitons to the mean curvature flow in $\mathbb{R}^{n+1}$

本文在 Spruck-Xiao 和 Derdziński 工作的基础上，证明了 n≥3 时 R^{n+1} 中完备 2-凸平移孤立子以及渐近于平均凸锥的完备 2-凸自扩张孤立子均为凸。

A note on the Gauss curvature flow

Using polar convex bodies and the $C_0$-bounds from Guan and Ni \cite{PL}, we obtain a uniform lower bound on the Gauss curvature of the normalized solution of the Gauss curvature flow without using Chow's Harnack inequality \cite{Ch2}.

Complete non-compact gradient Ricci solitons with nonnegative Ricci curvature

In this paper, we give a delay estimate of scalar curvature for a complete non-compact expanding (or steady) gradient Ricci soliton with nonnegative Ricci curvature. As an application, we prove that any complete non-compact expanding (or steady) gradient Kähler-Ricci solitons with positively pinched Ricci curvature should be Ricci flat. The result answers a question in case of Kähler-Ricci solitons proposed by Chow, Lu and Ni in a book.

单位球面乘积上的一个给定Gauss-Kronecker曲率问题

A prescribed Gauss-Kronecker curvature problem on the product of unit spheres

本文讨论了单位球面乘积上的一个给定Gauss-Kronecker曲率问题。

Spectral properties and rigidity for self-expanding solutions of the mean curvature flows

In this paper, we study self-expanders for mean curvature flows. First we show the discreteness of the spectrum of the drifted Laplacian on them. Next we give a universal lower bound of the bottom of the spectrum of the drifted Laplacian and prove that this lower bound is achieved if and only if the self-expander is the Euclidian subspace through the origin. Further, for self-expanders of codimension $1$, we prove an inequality between the bottom of the spectrum of the drifted Laplacian and the bottom of the spectrum of weighted stability operator and that the hyperplane through the origin is …

Nonparametric hypersurfaces moving by powers of Gauss curvature

We study asymptotic behavior of nonparametric hypersurfaces moving by $α$ powers of Gauss curvature $α> 1/n$. Our work generalizes the results of V. Oliker [Oli91] for $α= 1$.

Minkowski空间中常Gauss曲率的整类空超曲面

Entire spacelike hypersurfaces of constant Gauss curvature in Minkowski space

证明了Minkowski空间中具有给定Gauss曲率的光滑整严格凸类空超曲面的存在性和稳定性。证明是基于障碍的建设和当地的先验估计。

外区域中具有给定Gauss曲率的超曲面

Hypersurfaces of Prescribed Gauss Curvature in Exterior Domains

证明了欧几里得空间中紧集的补空间中具有给定高斯曲率的凸超曲面的一个存在性定理。

具有零高斯曲率的超曲面的三种Bernstein型定理

Three Bernstein type theorems for hypersurfaces with zero Gaussian curvature

本文研究了具有零高斯曲率的凸超曲面在欧几里得和闵氏背景下，证明了在特定条件下这些超曲面必须为超平面，并通过例子说明了额外条件的重要性。

An asymptotically cusped three dimensional expanding gradient Ricci soliton

We construct an expanding gradient Ricci soliton in dimension three over the topological manifold R x T^2 (the product of a line and a torus) that aproaches asymptotically a constant curvature cusp at one end, and a flat manifold on the other end. We prove that this is the only gradient soliton with this topology, provided the curvature is negatively pinched, -1/4 < sec < 0, at the time-zero manifold (normalizing the soliton to be born at time -1).

A Gauss curvature flow related to the Orlicz-Aleksandrov problem

In this paper we first obtain the existence of smooth solutions to Orlicz-Aleksandrov problem via a Gauss-like curvature flow.

有界宽度的古代高斯曲率流

Ancient Gauss curvature flows of bounded width

本文构造了高斯曲率流中煎饼状古代紧致解及$\alpha$-高斯曲率流中香肠状古代紧致解，后者渐近于圆柱体。

Self-expanders of the mean curvature flow

We study self-expanding solutions $M^m\subset\mathbb{R}^{n}$ of the mean curvature flow. One of our main results is, that complete mean convex self-expanding hypersurfaces are products of self-expanding curves and flat subspaces, if and only if the function $|A|^2/|H|^2$ attains a local maximum, where $A$ denotes the second fundamental form and $H$ the mean curvature vector of $M$. If the pricipal normal $ξ=H/|H|$ is parallel in the normal bundle, then a similar result holds in higher codimension for the function $|A^ξ|^2/|H|^2$, where $A^ξ$ is the second fundamental form with respect to $ξ$. …

Isometric Immersions via Compensated Compactness for Slowly Decaying Negative Gauss Curvature and Rough Data

In this paper the method of compensated compactness is applied to the problem of isometric immersion of a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. Previous applications of the method to this problem have required decay of order $t^{-4}$ in the Gauss curvature. Here we show that the decay of Hong $t^{-2-δ/2}$ where $δ\in(0,4)$ suffices.

Flow by powers of the Gauss curvature in space forms

In this paper, we prove that convex hypersurfaces under the flow by powers $α>0$ of the Gauss curvature in space forms $\mathbb{N}^{n+1}(κ)$ of constant sectional curvature $κ$ $(κ=\pm 1)$ contract to a point in finite time $T^*$. Moreover, convex hypersurfaces under the flow by power $α>\frac{1}{n+2}$ of the Gauss curvature converge (after rescaling) to a limit which is the geodesic sphere in $\mathbb{N}^{n+1}(κ)$. This extends the known results in Euclidean space to space forms.

Surfaces moving by powers of Gauss curvature

We prove that strictly convex surfaces moving by $K^{α/2}$ become spherical as they contract to points, provided $α$ lies in the range $[1,2]$. In the process we provide a natural candidate for a curvature pinching quantity for surfaces moving by arbitrary functions of curvature, by finding a quantity conserved by the reaction terms in the evolution of curvature.

Minkowski空间中具有给定Gauss曲率的整类空超曲面

Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space

研究Minkowski空间中具有正常数或给定Gauss曲率的类空凸超曲面（K-超曲面）。我们的主要目的是研究相关的Monge-Ampere方程的整体解以及有界区域上的狄利克雷问题。

Isometric Immersion of Complete Surfaces with Slowly Decaying Negative Gauss Curvature

The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is considered in this paper through the Gauss-Codazzi system for the second fundamental forms. It is shown that if the Gauss curvature satisfies an integrability condition, the surface has a global smooth isometric immersion in the three-dimensional Euclidean space even if the Gauss curvature decays very slowly at infinity. The new idea of the proof is based o…

Inverse Gauss curvature flows with free boundaries in a cone

We consider strictly convex hypersurfaces with the boundary which meets a strictly convex cone perpendicularly. We prove that if these hypersurfaces expand inside this cone, driven by the power of the Gauss curvature, then the evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece of round sphere after rescaling.

A generalized Gauss curvature flow related to the Orlicz-Minkowski problem

In this paper a generalized Gauss curvature flow about a convex hypersurface in the Euclidean $n$-space is studied. This flow is closely related to the Orlicz-Minkowski problem, which involves Gauss curvature and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As a byproduct, two existence results of solutions to the even Orlicz-Minkowski problem are obtained, one of which improves the known result.

Weak stability of Ricci expanders with positive curvature operator

We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity.

具有Neumann边界条件的Gauss曲率流的平移解

Translating solutions for Gauss curvature flows with Neumann boundary conditions

我们考虑了由非参数对数高斯曲率流演化的严格凸超曲面，它们服从Neumann边界条件。解决方案示出了顺利收敛到超曲面移动平移。特别地，对于有界区域，我们证明了具有指定法向导数的凸函数满足一致振动估计。

Uniformly Area Expanding Flows in Spacetimes

The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse mean curvature vector flow equation lacks a general existence theory. Our main contribution is proving that there exist infinitely many spacetimes, not necessarily spherically symmetric or static, that admit smooth global solutions to inverse mean curvature vector flow. Prior to our work, such solutions were only known in spherically symmetric and static spacetimes. The technique used in this thes…

On the Local Isometric Embedding in $R^3$ of Surfaces with Zero Sets of Gaussian Curvature Forming Cusp Domains

We study the problem of isometrically embedding a two-dimensional Riemannian manifold into Euclidean three-space. It is shown that if Gaussian curvature vanishes to finite order and its zero set consists of two smooth curves tangent at a point, then local sufficiently smooth isometric embedding exists.

Centro-affine curvature flows on centrally symmetric convex curves

We consider two types of $p$-centro affine flows on smooth, centrally symmetric, closed convex planar curves, $p$-contracting, respectively, $p$-expanding. Here $p$ is an arbitrary real number greater than 1. We show that, under any $p$-contracting flow, the evolving curves shrink to a point in finite time and the only homothetic solutions of the flow are ellipses centered at the origin. Furthermore, the normalized curves with enclosed area $π$ converge, in the Hausdorff metric, to the unit circle modulo SL(2). As a $p$-expanding flow is, in a certain way, dual to a contracting one, we prove t…

带收缩障碍的高斯曲率流

Gauss curvature flow with shrinking obstacle

该研究针对带障碍约束的高斯曲率幂次流，在全维度正幂次下证得解的最优曲率界与长时间存在性，并在一致厚度假设下得到自由边界的$C^1$正则性，丰富了带约束曲率流理论。

Convergence of Gauss curvature flows to translating solitons

We address the asymptotic behavior of the $α$-Gauss curvature flow, for $α>1/2$, with initial data a complete non-compact convex hypersurface which is contained in a cylinder of bounded cross section. We show that the flow converges, as $t \to +\infty$, locally smoothly to a translating soliton which is uniquely determined by the asymptotic cylinder of the initial hypersurface.

On existence of hypersurfaces translating by powers of Gauss curvature

In this paper we construct complete convex hypersurfaces in $\mathbb R^{n+1}$ which translate under the flow by powers $α\in (0, \frac1{n+2})$ of the Gauss curvature. The level set of each solution is asymptotic to a shrinking soliton for the flow by power $\frac α{1-α}$ of the Gauss curvature in $\mathbb R^n$. For example, our construction reveals the existence of translators whose level set converges to the sphere, simplex, hypercube and so on. The translating solitons exist as a family whose parameters correspond to Jacobi fields, solutions to linearized equation around the asymptotic profi…

弱锥形膨胀梯度 Ricci 孤立子

On the weakly conical expanding gradient Ricci solitons

本文通过在球面上构造不同极限行为的度量序列，并结合 Deruelle 的工作与局部化最大值原理，构造了具有正曲率和奇异曲率衰减的膨胀梯度 Ricci 孤立子，分别解答了 Chow-Lu-Ni 与 Cao-Liu 的问题。

平面网络流的自相似展开解

Self similar expanding solutions of the planar network flow

本文证明了平面网络上曲率流的自相似展开解的存在性，其中初始构形是任意多个在原点相交的半直线。这概括了施努尔和舒尔茨最近的工作，其中对待三个半线的情况。有多个解决方案，这些是参数化的组合对象，即斯坦纳树相对于一个完整的负弯曲度量的单位球上的跨度$k$指定点的边界上的无穷大。我们还提供了一个尖锐的制定这些解决方案的规律性在$t=0$。

Entire self-similar solutions to Lagrangian Mean curvature flow

We consider self-similar solutions to mean curvature evolution of entire Lagrangian graphs. When the Hessian of the potential function $u$ has eigenvalues strictly uniformly between -1 and 1, we show that on the potential level all the shrinking solitons are quadratic polynomials while the expanding solitons are in one-to-one correspondence to functions of homogenous of degree 2 with the Hessian bound. We also show that if the initial potential function is cone-like at infinity then the scaled flow converges to an expanding soliton as time goes to infinity.

Rotational symmetry of non negatively curved expanding gradient Ricci solitons

Let $(M^n,g,\nabla f)$, $n\geq 3$, be an expanding gradient Ricci soliton with nonnegative sectional curvature whose asymptotic cone is isometric to $C(\mathbb{S}^{n-1}(c))$ where $\mathbb{S}^{n-1}(c)$ is the standard $(n-1)$-sphere of curvature $1/c^2$, with $c\in(0,1)$. We prove that if the convergence to the asymptotic cone is smooth, $(M^n,g,\nabla f)$ is rotationally symmetric. This is the expanding analogue of the Perelman conjecture on the Bryant soliton and this work is based on the proof by Brendle \cite{Bre-Rot-3d}. This has also been proved recently by Chodosh \cite{Cho-EGS}.

On Gaussian curvatures and singularities of Gauss maps of cuspidal edges

We show relation between sign of Gaussian curvature of cuspidal edge and geometric invariants through types of singularities of Gauss map. Moreover, we define and characterize positivity/negativity of cusps of Gauss maps by geometric invariants of cuspidal edges, and show relation between sign of cusps and of the Gaussian curvature.

具有常数量曲率和零高斯曲率的$\mathbb{S}^5$中极小超曲面是全测地的

Minimal hypersurfaces in $\mathbb{S}^5$ with constant scalar curvature and zero Gauss curvature are totally geodesic

本文证明五维球面中常数量曲率且零高斯曲率的闭极小超曲面必为全测地超曲面。

具有凸高斯像的高余维平均曲率流古老解的曲率估计

Curvature estimates of ancient solutions to the mean curvature flow of higher codimension with convex Gauss image

本文通过进行精细的曲率估计，在各种高斯像限制条件下，证明了高余维平均曲率流完备非紧古老解的更好刚性定理。

$\mathbb R^m$中Gauss映射省略超平面集的极小曲面的Gauss曲率估计

An estimate for the Gauss curvature of minimal surfaces in $\mathbb R^m$ whose Gauss map omits a set of hyperplanes

对于${\mathbb R}^m$中的极小曲面，当其高斯映射省略了${\mathbb P}^{m-1}({\mathbb C})$中的多个$m(m+1)/2$超平面时，给出了其高斯曲率的一个估计.